Resonances in the Two-Centers Coulomb Systems arXiv:1409.5580v3

Resonances in the Two-Centers Coulomb Systems

arXiv:1409.5580v3 [math-ph] 6 Oct 2016

Marcello Seri∗, Andreas Knauf†, Mirko Degli Esposti‡ and Thierry Jecko§

We investigate the existence of resonances for two-centers Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalised complex eigenvalues of a non-selfadjoint deformation of the two-centers Schr¨ odinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analysed by means of perturbation theory and numerical methods. Mathematics Subject Classification: 34E20, 34F15, 35P15, 81U05, 81V55

Contents 1. Introduction 2. The 2.1. 2.2. 2.3. 2.4.

two-centers system on L2 (R2 ) The two-centers Coulomb system Elliptic coordinates . . . . . . . Classical results . . . . . . . . . Separation in elliptic coordinates

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3. Spectrum of the angular operator and its analytic continuation

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Department of Mathematics and Statistics University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX (UK), [email protected] † Department of Mathematics, Friedrich-Alexander-University Erlangen-Nuremberg, Cauerstr. 11, D-91058 Erlangen, Germany, [email protected] ‡ Dipartimento di Matematica, Universit` a di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy, [email protected] § D´epartement de Math´ematiques, Universit´e de Cergy-Pontoise, Site de Saint Martin, 2, avenue Adolphe Chauvin, F-95000 Cergy-Pontoise, France, [email protected]

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4. Asymptotic behaviour of solutions of the radial Schr¨ odinger equation and their analytic extensions 4.1. Decomposition into long and short range . . . . . . . . . . . . . . . . . . . . 4.2. Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Generalised eigenfunctions, Green’s function and the scattering matrix . . . .

13 14 22 25

5. Formal partial wave expansion of the Green’s function

28

6. Resonances for the two-centers problem 6.1. Definition of the resonances . . . . . . 6.2. Computation of the resonances of Kξ 6.3. Resonances for Z− = 0 . . . . . . . . 6.4. Resonances for Z− > 0 . . . . . . . . 6.5. High energy estimates . . . . . . . . .

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7. Numerical investigations

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8. The two-center problem in 3D and the n-center problem

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A. Generalised Pr¨ ufer transformation in the semi-classical limit

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References

46

1. Introduction Our work concerns the study of the quantum mechanical two-fixed-centers Coulomb systems in two dimensions. The two-dimensional restriction of the two-centers problem arises naturally in the analysis of the three-dimensional problem and, as described in [48], it is essential to be able to analyse that case. Since three centuries the two-centers Coulombic systems have been studied, from a classical and later also from a quantum mechanical point of view, starting from pioneering works of Euler, Jacobi [28] and Pauli [43] and going on until the recent years. For an historical overview we refer the reader to [48]. The interest for the quantum mechanical version of the problem comes mainly from molecular physics. Indeed it defines the simplest model for one-electron diatomic molecules (e.g. ++ ) and a first approximation of diatomic molecules in the Bornthe ions H+ 2 and He H Oppenheimer representation. In fact many of the results in the literature are related to the hard problem of finding algorithms to obtain good numerical approximations of the discrete spectrum and of the scattering waves [22, 23, 35, 36, 47], and to the asymptotic analysis of spectral properties in the very small or very large center distance [10, 15, 21, 31]. In contrast, really little is known on the regularity of the solutions with respect to the parameters of the system [52] and even less on the problem of resonances. Quantum resonances are a key notion of quantum physics: roughly speaking these are

2

scattering states (i.e. states of the essential spectrum) that for long time behave like bound states (i.e. eigenfunctions). They are usually defined as poles of a meromorphic function, but note that there is no consensus on their definition and their study [62]. On the other hand, it is known that many of their definitions coincide in some settings [26] and that their existence is related to the presence of some classical orbits “trapped” by the potential. If a quantum systems has a potential presenting a positive local minimum above its upper limit at infinity, for example, it is usually possible to find quantum resonances, called shape resonances. These are related to the classical bounded trajectories around the local minimum [27]. These are not the only possible ones: it has been proven in [7, 8, 20, 50] that there can be resonances generated by closed hyperbolic trajectories or by a non-degenerate maximum of the potential. The main difference is that the shape resonances appear to be localised much closer to the real axis with respect to these last ones. Even the presence or absence of these resonances is strictly related to the classical dynamics. In fact it is possible to use some classical estimates, called non-trapping conditions, to prove the existence of resonance free regions (see for example [6, 38, 39]). A major shortcoming of the actual theory of resonances is that the existence and localisation results require the potentials to be smooth or analytic everywhere, with the exception of few results concerning non-existence [38, 39] or restricting to centrally symmetric cases [3]. In this sense, the two-centers problem represents a very good test field. In fact, it is not centrally symmetric but presents still enough symmetries to be separated (see Theorem 2.7). This allows us to shift most of the analysis from the theory of PDEs with singular potentials to the theory of ODEs, simpler and more explicitly accessible. Moreover, the two-centers models present all the previously cited classical features related to the existence of resonances: the non-trapping condition fails to hold [11], there are closed hyperbolic trajectories with positive energies [32, 49] and there is a family of bounded trajectories with positive energies [49]. At the same time, the energy ranges corresponding to the closed hyperbolic trajectories and to the bounded ones are explicitly known [49]. In general the relation between different definitions of resonances is not fully understood, even for smooth symbols. In this work we define a notion of resonances for the two-centers Coulomb system. These are defined as poles of the meromorphic extension of the Green’s functions of the separated equations. We then show how to approximate them in different semiclassical energy regimes. These approximations lead to strong evidence that relates the energies of the resonances far from the real axis (i.e. not-exponentially close to it w.r.t. the semiclassical parameter) to that of the closed hyperbolic trajectories. Our work is strongly inspired by [3] but we treat a more interesting situation since the scattering by two nuclei is richer than the one by one nucleus. We get similar results as in [3], except for the expansion of the Green function in partial waves. In [3], the latter can be justified thanks to a special property of spherical harmonics. We did not succeed in proving it in our context (and this would be an important result). This explains why we did not completely connect our definition of resonances to usual ones. Compared to other results on resonances, we provide quite precise informations in an usually unpleasant context since our potential (as in [3]) contains Coulomb singularities.

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Except for some results in Section 6.2, our main contributions are not of semi-classical nature in contrast to those in [6, 7, 8, 50]. The structure of the paper is as follows. In Section 2 we introduce the two-centers problem both in its classical and quantum mechanical formulation. We describe its main properties and the separation of the differential equation associated to the operator into radial and angular equations. In Section 3 we describe the spectrum of the operator obtained from the angular differential equation and the properties of its analytic continuation. In Section 4 we focus on the spectrum of the operator obtained from the radial differential equation and the analytic continuation of its resolvent. This is done constructing explicitly two linearly independent solutions with prescribed asymptotic behaviour. They mimic the incoming and outgoing waves of scattering theory, in fact we will use them to construct the Jost functions, and consequently define and analyse the Green’s function and the scattering matrix. The main results are contained in Theorem 4.5 and Theorem 4.14 and their corollaries. In particular they provide the key ingredients to define the Jost functions and their analytic continuation in Corollary 4.19. In Theorem 4.5 is proven the existence and uniqueness of the incoming and outgoing waves for real and complex values of the parameters. In Theorem 4.14, it is shown that these solutions admit an analytic continuation across the positive real axis into the second Riemann sheet. In Section 5 we explain how the resolvent of the two-centers system relates to the angular and radial operators. In Section 6 we apply the theory developed for the angular and radial operators to the objects described in Section 5. Here we define the resonances for the two-centers problem (see (6.2)) and analyse some of their properties. The rest of the section is devoted to the computation of approximated values of the resonances in different semiclassical energy regimes, see in particular (6.9), (6.18) and (6.22). In Section 7 we use the approximations obtained in the previous section to compute the resonances and study their relationship with the structure of the underlying classical systems. The numerics strongly support the relation between the resonances that we’ve found and the classical closed hyperbolic trajectories. In Section 8 we make some additional comments relating our results for the planar twocenters problem to the three-dimensional one and to the n-centers problem. In the Appendix A we describe how to modify the generalised Pr¨ ufer transformation in the semi-classical limit to get precise high-energy estimates. These results are needed for the high-energy approximation obtained in Section 6.5. Notation. In this article N = {1, 2, 3, . . .}, R∗ := R \ {0}.

2. The two-centers system on L2 (R2 ) 2.1. The two-centers Coulomb system We consider the operator in L2 (R2 ), given by H := −h2 ∆ + V (q)

with V (q) :=

4

−Z1 −Z2 + , |q − s1 | |q − s2 |

(2.1)

where h > 0 is a small parameter. This describes the motion of an electron in the field of two nuclei of charges Zi ∈ R∗ = R \ {0}, fixed at positions s1 6= s2 ∈ R2 , taking into account only the electrostatic force. By the unitary realisation U f (x) := | det A|−1/2 f (Ax + b) of an affinity of R2 we assume that the two centers are at s1 := a := ( 10 ) and s2 := −a. Remarks 2.1. • Notice that if we set Z1 = Z2 > 0 in the operator in (2.1), we get the Schr¨odinger operator for the simply ionized hydrogen molecule H+ 2 [10, 15, 53], whereas for Z1 = −Z2 it describes an electron moving in the field of a proton and an anti-proton [21, 31]. Another example covered by this model is the doubly charged helium-hydride molecular ion He H++ , with Z1 = 2Z2 > 0, see [60]. • Even if (2.1) does not directly describe the interactions in molecules, it is related to the study of scattering theory for such systems. In Example 1.3 in [11], the scattering of a heavy particle by a molecule is partially studied and, thanks to a natural physical assumption, the Hamiltonian of the heavy particle is given by (2.1) plus an additional potential correction. In the paper [29], scattering cross sections for diatomic molecules are estimated in a semi-classical regime related to the Born-Oppenheimer approximation. A Schr¨ odinger operator of the type (2.1) enters in the computations as an effective Hamiltonian for the scattering process. ♦

2.2. Elliptic coordinates The restriction to the rectangle M := (0, ∞) × (−π, π) of the map  
cosh(ξ) cos(η) ξ G : R2 → R2 , → 7 η sinh(ξ) sin(η)

(2.2)

defines a C ∞ diffeomorphism G : M → G(M )

(2.3)

whose image G(M ) = R2 \ (R × {0}) is dense in R2 . Moreover it defines a change of coordinates from q ∈ R2 to (ξ, η) ∈ M . These new coordinates are called elliptic coordinates. Remarks 2.2. 1. In the (q1 , q2 )-plane the curves ξ = c are ellipses with foci at ±a, while the curves η = c are confocal half hyperbolas, see Figure 2.1. 2. The Jacobian determinant of G equals F (ξ, η) := det(DG(ξ, η)) = sinh2 (ξ) + sin2 (η) = cosh2 (ξ) − cos2 (η).

(2.4)

Thus the coordinate change (2.2) is degenerate at the points (ξ, η) ∈ {0} × {0, ±π} in M . For ξ = 0 the η coordinate parametrizes the q1 -axis interval between the two centers. For η = 0 (η = ±π) the ξ coordinate parametrizes the positive (negative) q1 -axis with |q1 | > 1. ♦

5

q2 1

-1

1

q1

-1

Figure 2.1: Elliptic coordinates.

2.3. Classical results The classical analogue of (2.1) is described by the Hamiltonian function on the cotangent bundle T ∗ Q2 of Q2 := R2 \ {±a} relative to the two-center potential given by: H : T ∗ Q2 → R

, H(p, q) :=

−Z1 −Z2 |p|2 + + . 2 |q − a| |q + a|

(2.5)

Lemma 2.3 (see e.g. [49]). Using G defined in (2.3), and Z± := Z2 ± Z1 , H is transformed by the elliptic coordinates into H ◦ (G−1 )∗ (pξ , pη , ξ, η) =


1 H1 (pξ , ξ) + H2 (pη , η) F (ξ, η)

(2.6)

where (G−1 )∗ : T ∗ M → T ∗ Q2 is the cotangential lift of G−1 , and H1 (pξ , ξ) :=

p2ξ 2

− Z+ cosh(ξ)

, H2 (pη , η) :=

p2η + Z− cos(η). 2

(2.7)

There are two functionally independent constants of motion H and L := H1 − cosh2 (ξ)H with values E and K respectively. Taken together, the constants of motion define a vector-valued function on the phase space of a Hamiltonian. We can study the structure of the preimages of this function (its level sets), in particular their topology. In the simplest case the level sets are mutually diffeomorphic manifolds. Definition 2.4. (see [1, Section 4.5]) Given two manifolds M, N , f ∈ C ∞ (M, N ) is called locally trivial at y0 ∈ N if there exists a neighborhood V ⊆ N of y0 such that f −1 (y) is a smooth submanifold of M for all y ∈ V and there there is a map g ∈ C ∞ (f −1 (V ), f −1 (y0 )) such that f × g : f −1 (V ) → V × f −1 (y0 ) is a diffeomorphism. The bifurcation set of f is the set B(f ) := {y0 ∈ N | f is not locally trivial at y0 }.

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Notice that if f is locally trivial, the restriction gf −1 (y) : f −1 (y) → f −1 (y0 ) is a diffeomorphism for every y ∈ V . Remark 2.5. The critical points of f lie in B(f ) (see [1, Prop. 4.5.1]), but the converse is true only in the case f is proper (i.e. it has compact preimages). ♦ Define the function on the phase space as follows (omitting a projection in the second component)   F := HξH◦G∗ : T ∗ Q2 → R2 , (2.8) where Hξ (pξ , ξ) := H1 (pξ , ξ) − cosh2 (ξ)E. Theorem 2.6 ([49]). Let (Z1 , Z2 ) ∈ R∗ × R∗ , then the bifurcation set of (2.8) for positive energies equals B (F) ∩ (R+ × R) = {(E, K) ∈ L | E ≥ 0 and K+ (E) ≤ K ≤ K− (E)} . Here L := L0 ∪ L1− ∪ L2− ∪ L3− ∪ L2+ ∪ L3+ ⊂ R2 with L1− := {K = Z− − E}, L2− := {K = −Z− − E}, 2 }, L3− := {4EK = Z−

L0 := {E = 0}, L2+ := {K = −Z+ − E}, 2 }, L3+ := {4EK = Z+

(2.9)

and K+ and K− are defined by  −∞,    K+ (E) := −(Z+ + E),    Z+2 ,

E>0   E ≤ min − Z2+ , 0 ,   0 ≥ E > min − Z2+ , 0

4E

K− (E) :=

( Z− − E, 2 Z− 4E ,

E≤ E>

Z− 2 Z− 2

.

The energies lying on the line L2+ are the ones associated with the closed hyperbolic trajectory bouncing between the two centers [49]. Moreover, for |Z+ | < Z− the n o set of energy parameters included in the region {E ≥ |Z+ | 3 0} ∩ (E, K) ∈ L+ E < 2 and contained between the curves L2+ and L1+ is somewhat special: on the configuration space they are associated with a family of bounded trajectories trapped near the attracting center [49].

2.4. Separation in elliptic coordinates The importance of the change of coordinate (2.3) for the quantum problem is clarified by the following well-known theorem (see e.g. [5]). Here we enlarge the domain of G to M .

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n o Theorem 2.7. Let u ∈ Ca (R2 ) := u ∈ C(R2 ) uR2 \{±a} is twice continuously differentiable . The eigenvalue equation
− h2 ∆ + V (q) u(q) = Eu(q), E ∈ R, transformed to prolate elliptic coordinates, separates with the ansatz u ◦ G(ξ, η) = f (ξ)g(η) into the decoupled system of ordinary differential equations  ( −h2 ∂ξ2 − Z+ cosh(ξ) − E cosh2 (ξ) + µ f (ξ) = 0
−h2 ∂η2 + Z− cos(η) + E cos2 (η) − µ g(η) = 0, where µ ∈ C is the separation constant,  2 f ∈ CN ([0, ∞)) := h ∈ C 2 ([0, ∞)) | h0 (0) = 0 , n o 2 g ∈ Cper ([−π, π]) := h ∈ C 2 ([−π, π]) | h(k) (−π) = h(k) (π) for k = 0, 1 and we have set Z± := Z2 ± Z1 and ∂α =

∂ ∂α .

Remark 2.8. Without loss we assume Z− ∈ [0, ∞) and Z+ ∈ R, Z+ 6= Z− , i.e. Z2 ≥ Z1 . ♦ Remark 2.9. Since G is a diffeomorphism and since F defined in (2.4) equals det(DG), the transformation to prolate elliptic coordinates (ξ, η) defines a unitary operator G : L2 (R2 , dq) → L2 (M, dχ) , with

dχ := F (ξ, η) dξ dη. ♦

Proof of 2.7. We set r1 := |q − s1 |, r2 := |q − s2 | and transform to elliptic coordinates. We have
2 2 r2,1 = (q1 ± 1)2 + q22 = cosh(ξ) ± cos(η) . Thus the distances from the centers equal r1 = cosh ξ − cos η

and r2 = cosh ξ + cos η.

For F (ξ, η) = sinh2 (ξ) + sin2 (η) = cosh2 (ξ) − cos2 (η) we obtain V ◦ G(ξ, η) = −

Z1 Z2 Z+ cosh(ξ) − Z− cos(η) − =− |q − a| |q + a| F (ξ, η)

and the Laplacian ∆ acts in elliptic coordinates as ∆G :=


1 ∂ξ2 + ∂η2 . F (ξ, η)

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(2.10)

With the ansatz u e(ξ, η) = f (ξ)g(η)

with

2 2 f ∈ CN ([0, ∞)) and g ∈ Cper ([−π, π])

the first equation separates and we obtain the decoupled system of ordinary differential equations − h2 ∂ξ2 f (ξ) + (Vξ (ξ) + µ) f (ξ) = 0 ,

− h2 ∂η2 g(η) + (Vη (η) − µ) g(η) = 0

(2.11)

where Vξ and Vη are the multiplication operators for the functions Vξ (ξ) := −Z+ cosh(ξ) − E cosh2 (ξ)

, Vη (η) := Z− cos(η) + E cos2 (η)

(2.12)

Remark 2.10. Here the separation constant µ plays the role of the spectral parameter in time independent Schr¨ odinger equations, and energy E the one of a coupling constant. ♦ Proposition 2.11. The operator H on L2 (R2 ) defined as in (2.1) is unitarily equivalent to the operator in L2 (M, dχ), given by HG := −h2 ∆G + VG

with

VG (ξ, η) := −

Z+ cosh(ξ) − Z− cos(η) . F (ξ, η)

HG has form core

G C0∞ (R2 ) = f ∈ C0∞ M | f (ξ, π) = f (ξ, −π) and ∂ξ f |ξ=0 = 0 . It admits a unique self-adjoint realisation with domain G(D(H)) with  1 D(H) := u ∈ L2 (R2 ) | V u ∈ L1loc (R2 ), u ∈ Hloc (R2 ), Hu ∈ L2 (R2 ) ,

(2.13)

where Hu is to be understood in distributional sense. Proof. It is well-known that H has a self-adjoint realisation on L2 (R2 ). The proof is based on the infinitesimal form boundedness of V w.r.t. ∆ [2, Theorem 3.2] and the KLMN Theorem [44, Theorem X.17]. In this way the operator is well-defined and has form domain H 1 (R2 ). Moreover its domain D(H) is given by (2.13), see [2, Theorem 3.2]. The domain of the unitarily transformed HG = GHG −1 is then transformed to G(D(H)). Finally C0∞ (R2 ) is a form core for the quadratic form associated to H, therefore it is unitarily transformed to a form core for the quadratic form associated to HG . See [46, Section VIII.6] for the definitions. The form of the operator is given by Theorem 2.7. It is natural at this point to move our point of view from the study of HG −E on L2 (M, dχ) to the study of the separable operator KE := Kξ ⊗ 1l + 1l ⊗ Kη acting on L2 (M, dξ dη) = L2 ([0, ∞), dξ) ⊗ L2 ([−π, π], dη). Here Kξ (h) := Kξ,E,h := −h2 ∂ξ2 − Z+ cosh(ξ) − E cosh2 (ξ), Kη (h) := Kη,E,h := −h2 ∂η2 + Z− cos(η) + E cos2 (η).

9

(2.14)

In fact, the separation reduces the problem to the study of two Sturm-Liouville equations (Kξ + µ)f (ξ) = 0

and

(Kη − µ)g(η) = 0.

(2.15)

Following the standard convention used in the literature, we will call the first equation radial equation and the second equation angular equation. For the proper boundary conditions on L2 ([0, ∞), dξ) respectively L2 ([−π, π], dη) they define essentially self-adjoint operators. More specifically the eigenvalue equation of Kη (h) is in the class of the so called Hill’s equation. In view of Proposition 2.11, we are interested in the 2π-periodic solutions of the equation, i.e. we look for g ∈ L2 ([−π, π], dη) such that g(−π) = g(π)

and g 0 (−π) = g 0 (π).

For Kξ (h) it is clear that 0 is a regular point, we will see later how to treat the singular point ∞ (we refer the reader to [61] for additional information concerning regular and singular points of Sturm-Liouville Problems). For what concerns the boundary conditions in 0, as suggested by Proposition 2.11 we will require f 0 (0) = 0.

(2.16)

The transformation needed to move from HG −E to (2.14) is obviously not unitary, as we are passing from a semibounded operator to a family of non-semibounded ones. On the other hand, their spectra are related, and we will study σ(HG ) by means of the spectra associated to (2.14).

3. Spectrum of the angular operator and its analytic continuation We now turn the attention to the second equation in (2.15), the angular equation. Let T := Tη (Z− , h, µ, E) := h−2 Kη (h) − h−2 E,

(3.1)

with parameters Z− ∈ R and E ∈ (0, ∞). With this definition, h2 [T ψ](η) = 0 denotes the eigenvalue equation for Kη . We start considering the simpler case of equal charges (Z− = 0). Then the eigenvalue equation [T ψ](η) = 0 is the Mathieu equation [T ψ](η) = −∂η2 ψ(η) −

2µ − E E ψ(η) + 2 2 cos(2η)ψ(η) = 0 2 2h 4h

(3.2)

with periodic boundary conditions in [−π, π]. We apply Floquet theory (see [16, 37, 41, 55]), using the fundamental matrix   2µ − E E f f F(λ, δ) := f10 f20 (π; λ, δ), λ := , δ := 2 , (3.3) 2 1 2 2h 4h built from the fundamental system of solutions η 7→ fi (η; λ, δ), with f1 (0; λ, δ) = 1 = f20 (0; λ, δ)

, f2 (0; λ, δ) = 0 = f10 (0; λ, δ)

10

(3.4)

(henceforth the prime 0 means the partial derivative w.r.t. the first variable). The potential V (η) := cos(2η) being even, it follows that all the 2π-periodic solutions must be either π-periodic or π-antiperiodic in [0, π] (or [−π, 0]). The structure of the periodic solutions and their eigenvalues for the Mathieu equation is well-understood (see [41, Chapter 2]): For each integer n ≥ 0 one finds two solutions cen (•; δ) and sen+1 (•; δ), called Mathieu Cosine and Mathieu Sine respectively, that have exactly n zeroes in (0, π) and that are π-periodic for even n and π-antiperiodic for odd n, − the corresponding eigenvalues being λ+ n (δ) and λn+1 (δ) respectively. For parameter values − + E ∈ R, δ ∈ (0, ∞) the λn and λn+1 are real and − + − + λ+ 0 < λ1 < λ1 < λ2 < λ2 < · · · .

The following facts are proved in [30, Chapter VII.3.3], [40, Chapter 2.4], [41, Chapter 2.2] and [58]. 1. The eigenvalues of the Mathieu operators are real-analytic functions in δ ∈ C, whose algebraic singularities all lie at non-real branch points. 2. They can be defined uniquely as functions λ± n (δ) of δ by introducing suitable cuts in the δ-plane. Moreover they admit an expansion in powers of δ with finite convergence radius rn such that lim inf n→∞ nrn2 ≥ C for some C > 0. 3. The number of branch points is countably infinite, and there are no finite limit points. 4. The operator T corresponding to (3.2) can be decomposed according to − − + L2 ([−π, π]) = L+ 0 ⊕ L1 ⊕ L0 ⊕ L1

where the superscripts ± denote respectively the sets of even and odd functions and where the subscripts 0 and 1 denote respectively the sets of functions symmetric and antisymmetric with respect to x = π/2. 5. The restrictions of T to the four subspaces L± 0/1 are self-adjoint and have only simple eigenvalues, as given by the following scheme: + + L+ 0 : λn , ψn ,

n = 0, 2, 4, 6, . . . ;

L+ 1 L− 0 L− 1

n = 1, 3, 5, . . . ;

: : :

+ λ+ n , ψn , − λ− n , ψn , − λ− n , ψn ,

n = 1, 3, 5, . . . ; n = 2, 4, 6, . . . .

6. All the eigenvalues in each of the four groups of the previous remark belong to the same analytic function, i.e. the eigenvalues in the same group lie on the same Riemann surface [41, 59]. 7. The eigenfunctions η 7→ ψn± (η) are themselves analytic functions of x and δ. For all n ∈ N they coincide with the Mathieu Cosine and the Mathieu Sine introduced above, − namely ψn+ ≡ cen and ψn+1 ≡ sen+1 (n ∈ N0 ). Despite the completeness and the clarity of perturbation theory for one-parameter analytic families of self-adjoint operators, the situation is much more intricate and much less complete in presence of more parameters. On the other hand we can use our restrictions on the parameters and the special symmetries of the potential to play in our favor.

11

For a general value of Z− , the eigenvalue equation is   Z− E 2µ − E 2 ψ(η) = 0, [T ψ](η) = −∂η ψ(η) + cos(η) + 2 cos(2η) − h2 2h 2h2

(3.5)

with periodic boundary conditions on [−π, π] and eigenvalue µ. Let us call λ :=

2µ − E , 2h2

γ1 :=

Z− , h2

γ2 :=

E . 2h2

(3.6)

Notice that the main difference between (3.5) and the Mathieu equation is that now the period of the potential is no more smaller than the length of the considered interval. Thus, in applying Floquet theory we do not anymore look for solutions which are (anti-)periodic under translation by π. Remark 3.1. By standard Sturm-Liouville theorems (see for instance [16, Theorems 2.3.1 and 3.1.2]) we know that for every choice of γ1 and γ2 the spectrum of Kη (h) is discrete, at most doubly degenerate and accumulates only at infinity. Anyhow it follows from [37, Theorem 7.10] using a change of variable that in this case there cannot be coexistence of 2π-periodic eigenfunctions for the same eigenvalue. Thus the spectrum is non-degenerate. ♦ It is proved in [54] that, for real-valued E and Z− , the eigenvalues of h−2 Kη (h) form a countably infinite set {λn (γ1 , γ2 , h)}n≥0 of transcendental real analytic (actually entire) functions of the parameters γ1 , γ2 ∈ R, so that in the (γ1 , γ2 , λ) space the sets  (γ1 , γ2 , λn (γ1 , γ2 )) | (γ1 , γ2 ) ∈ R2 define a countably infinite number of uniquely defined real-analytic surfaces. We can apply analytic perturbation theory [30, Chapter VII] to T (β) := T + β(1 + cos(2η)) where T is defined in (3.1) and β is assumed to be defined by h and some real parameter Eim as follows √ Eim β(Eim , h) := i 2 (with i = −1, h ∈ (0, ∞)). 2h Therefore T (β) is merely (3.5) with complex E. It is evident that T (β) defines a selfadjoint analytic family of type (A) in the sense of Kato. Therefore [30, Chapter VII] each λn (γ1 , γ2 , β) admits an analytic extension on the complex plane around each real E that can be expanded as a series in β = iEim /2h2 with an n-dependent convergence radius ρn . Remark 3.1 concerning the simplicity of the spectrum and the construction described at points 4. and 5. on page 11 is still valid. Therefore we may continue to regard each eigenvalue as simple restricted on its proper subspace and consider the lower bound of the convergence radius in terms of the eigenvalues’ spacing in the proper subspace. These distances are known to be at least of order n, in the sense that there exists C > 0 such that lim inf n→∞ nth-distance ≥ C, see [30, Chapter VII.2.4]. n

12

In the particular case considered, we can use the ansatz given by [37, Theorem 1.1] to bound the distance between the periodic solutions with a boundary-value problem. To this end we can use the discussion of [57, Section 5] and apply it to our case to obtain the following rough estimate, generalizing point (2) on page 11. Theorem 3.2. Let E > 2|Z− |. Then the convergence radii ρD,N corresponding to (3.5) n with Dirichlet (resp. Neumann) boundary conditions satisfy lim inf n→∞

ρD,N 6 n ≥ 2 n 13

Proof. In [57], Section 5, it is shown that a result like our Theorem 3.2 holds for the Mathieu equation (see [57, Theorem 5.1]). This is a particular case of a more general theorem on the quadratic growth of the convergence radii for the eigenvalues of a big family of differential equations (see [57, Theorem 3.4]). To apply [57, Theorem 3.4] and obtain the theorem for the Mathieu equation, it is enough to check the assumptions and use the estimates obtained there to get the constants in the growth rate. This check relies on some crude estimates on incomplete elliptic integrals and on the potential that can be used also for our problem. Indeed, we can replace the estimate |2 cos(2z)| ≤ 2 cosh(2=z) for the Mathieu potential by a corresponding estimate for cos(2x) + Z2E− cos(x): if E  |2Z− |, then Z − cos(2z) + 2 cos(z) ≤ 2 cosh(2=z). E Then, the constants in the proof of [57, Theorem 5.1], would coincide with the constants 2 obtained for our potential: R = 2 cosh(2δ), R0 = 2, U 2 = π16 + δ 2 (notation from [57, Section 5]). And choosing δ = 21 one can check that the assumptions of [57, Theorem 3.4] 6 are satisfied and the growth constant is 13 also in this case. Remark 3.3. As for [57, Theorem 5.1], we used very crude estimates. The constants, and in particular the lower bound for the growth rate, are far from being optimal also in this case and could be improved following the enhancements presented in [58]. Remark 3.4. We expect that Theorem 3.2 still holds true for 0 < E ≤ 2|Z− |.

♦

4. Asymptotic behaviour of solutions of the radial Schr¨ odinger equation and their analytic extensions The general estimates that we develop in this section are needed in order to justify the formal step in the separation of variables and the construction of the Green’s functions. We proceed with a philosophy close to the one of [3]. With the substitution E = k 2 of its parameter, the radial equation in (2.15) takes the form
v 00 (ξ, k) + h−2 k 2 cosh2 (ξ) + Z+ cosh(ξ) − µ v(ξ, k) = 0 (4.1)

13

where ξ > 0, h > 0 and k ∈ C are arbitrary. Now for l ∈ N we set µ := µl , the l-th eigenvalue of Kη (counted in ascending order for real parameters and then extended analytically). We assume w.l.o.g. that h = 1, since h can be absorbed in the other parameters. We will be interested in the solutions v± (ξ, k) := v± (ξ, k, µ) of (4.1) which decay as ξ → ∞ for k in the upper, resp. lower, half-plane C± = {k ∈ C | =(k) ≶ 0}. We call them, following [3] “outgoing”, resp. “incoming”, and we will make a specific choice of such a family of solutions by fixing the behaviour of v± (ξ, k) as ξ → ∞. We want to construct a phase function that is an approximate solution of the eikonal equation for the Schr¨ odinger equation (4.1), that is characterized by a particular asymptotic behaviour and that is analytic in k. We would like to consider something of the form Z ξq φ(ξ, k) ∼ k 2 cosh2 (t) + Z+ cosh(t) − µ dt, (4.2) 0

but this gives a well-defined analytic function only for |k|2 > |Z+ − µ|. For our analysis it will be essential that the phase function is analytic in k ∈ C \ {0}. To construct it we reconsider the previous ansatz and perform a change of variables. If we call τ = sinh(t), the above equation is transformed into Z sinh(ξ) p µ Z+ . (4.3) k 2 − q(τ ) dτ with q(τ ) := −√ φ(ξ, k) ∼ 2 1+τ 1 + τ2 0 If we call r = sinh(ξ), we may consider the map r 7→ φ(arcsinh(r), k) to be the phase function of a long-range potential, asymptotic to r 7→ kr as r → ∞, (see (4.7) for a more precise statement), plus a short-range perturbation.

4.1. Decomposition into long and short range To construct the phase function φ, we introduce an appropriate decomposition of the potential q into short and long range parts. Let j ∈ N. We define lj , sj ∈ (0, ∞) → R by sj (τ ) := q(τ ) − lj (τ ) and lj (τ ) := −χ(τ ) √

Z+ , 1 + τ2

(4.4)

where χ(τ ) = 1 if Z+ ≥ 0 and otherwise is defined as follows: χ ∈ Cc∞ ((0, ∞); [0, 1]) such that χ(τ ) = 0 if τ ≤ j|Z+ | and χ(τ ) = 1 if τ ≥ j|Z+ | + 1. Note that sj (τ ) ∈ L1 ((0, ∞)), lj ∈ C 2 ((0, ∞)), sup lj (τ ) ≤ 1/j

and lj (τ ) = − √

τ >0

Z+ for τ > Rj , 1 + τ2

for Rj := j|Z+ | + 1. Let Ωj := {k ∈ C | |k|2 > 1/j} and φj ∈ (0, ∞) × Ωj → C, defined by Z φj (ξ, k) :=

sinh(ξ) q

k 2 − lj (τ ) dτ.

0

14

(4.5)

Here we have taken the principal branch of the square root, i.e. the uniquely determined √ analytic branch of z that maps (0, ∞) into itself. Note that φj (ξ, ·) is analytic in Ωj and φj (·, k) ∈ C 2 ((0, ∞)). Furthermore, for k ∈ Ωj , φj (·, k) satisfies the eikonal equation |∂ξ ψ(ξ)|2 = k 2 − lj (sinh(ξ))

(4.6)

on (0, ∞). Theorem 4.1. Let D := {(ξ, k) ∈ (0, ∞) × C \ {0} | sinh(ξ) ≥ |k −2 Z+ | + 1}. There exist a function φ : D → C satisfying the following properties: 1. For all (ξ, k) ∈ D, φ(ξ, −k) = −φ(ξ, k). 2. For all j ∈ N, the restriction of φ − φj to (Rj , ∞) × Ωj doesn’t depend on ξ and is an analytic function of k. 3. For all k ∈ C \ {0}, φ(ξ, ·) is analytic on each Ωj , for j ∈ N such that sinh(ξ) > Rj . 4. For all k ∈ C \ {0}, φ(·, k) ∈ C 2 ((0, ∞)) and satisfies the eikonal equation (4.6) on (Rj , ∞) where j is the integer part of |k 2 |−1 . The theorem follows from the construction above with the same proof as [3, Proposition 2.1]. Remark 4.2. The phase function φ defined in the previous theorem is not unique. This is, however, immaterial for our purposes. In fact, our main concern is to have a controlled behaviour, as ξ → ∞ (see Proposition 4.3) and good analyticity properties in order to identify the two (unique) waves v± for a wide range of parameters. Henceforth we will refer to the φ(ξ, k) defined in Theorem 4.1 as a global phase function. Proposition 4.3. The global phase function φ(ξ, k) has the asymptotic behaviour given by φ(ξ, k) = k sinh(ξ) +


Z+ k ξ + O(1) = eξ 1 + o(1) 2k 2

as ξ → ∞.

(4.7)

µ Remark 4.4. In the proposition the term s(τ ) := 1+τ 2 has been dropped out. In fact it belongs to the short range component sj of (4.4) and choosing in (4.4) a different decomposition of q(τ ) into a short-range and long-range part, keeping l(ξ) fixed near infinity, modifies φ(ξ, k) by an analytic function of k alone. ♦

Proof. Without losing generality we can suppose |k| > |Z+ | and consider the simplified phase function s Z ξ Z ξq Z+ dt (4.8) φ(ξ, k) := k 2 cosh2 (t) + Z+ cosh(t) dt = k cosh(t) 1 + 2 k cosh(t) 0 0

15

as ξ → ∞: Z φ(ξ, k) = k

ξ

 cosh(t) 1 +

0

= k sinh(ξ) +


Z+ + O k −2 cosh−2 (t) 2 2k cosh(t)

 dt

Z+ ξ + O(1), 2k

Writing sinh(ξ) = (ex − e−x )/2 and collecting the growing term we have the thesis. The Liouville-Green Theorem [17, Corollary 2.2.1] guarantees that for each k ∈ C there exist two linearly independent solutions of (4.1) whose asymptotics as ξ → ∞ is given by

y1,2 (ξ) = √ 01 exp ±iφ(ξ, k) 1 + o(1) for ξ → ∞. φ (ξ,k)

In particular, it follows from the asymptotic estimate of Proposition 4.3 that (4.1) must be in the Limit Point Case at infinity (more precisely Case I of [9, Theorem 2.1]) if we set r(x) := cosh2 (x), p := 1 and λ := k 2 . In what follows we investigate the regularity of the solutions with respect to ξ and k. Theorem 4.5 (Outgoing and incoming solutions). For each k ∈ C \ {0}, equation (4.1) has unique solutions v± (ξ, k) verifying the asymptotic relation √

ξ v± (ξ, k) = 2e− 2 exp ± iφ(ξ, k) 1 + o(1) as ξ → ∞. (4.9) (4.9) holds uniformly in any truncated cone Λ± (η, δ) := {k ∈ C \ {0} | η ≤ arg(±k) ≤ π − η, |k| ≥ δ}

with η ≥ 0, δ > 0.

The family of solutions k 7→ v± (ξ, k) defined by (4.9) is analytic in the half planes k ∈ C± pointwise in ξ, and extends continuously to k ∈ C± \ {0}. 0

-1

1

1

1

��(�)

Λ + (η , δ )

0

0

Λ - (η , δ ) -1

-1 0

-1

1

��(�) Figure 4.1: Cones Λ± for η = 1/3 and δ = 1/2.

16

Remark 4.6. (1) and the uniqueness of Theorem 4.5 imply that v+ (ξ, k) = v− (ξ, −k). In particular it suffices to consider v+ . ♦ Proof. In view of Theorem 4.1 and the subsequent remark, we can reduce the proof to the case where the phase function φ is given by (4.8) for ξ > 0 and |k|2 > |Z+ |. We call φ a local phase function. Let  V± (ξ, k) :=

k ∂ξ φ(ξ, k)

1 2

e±iφ(ξ,k)

(4.10)

define the approximate solutions of (4.1). For |k| ≥ δ the function V± satisfies the comparison equation
V±00 (ξ, k) + k 2 cosh2 (ξ) + Z+ cosh(ξ) + 12 Sφ(ξ, k) V± (ξ, k) = 0

(4.11)

where Sφ denotes the Schwarzian derivative φ000 3 Sφ = 0 − φ 2



φ00 φ0

2 (4.12)

w.r.t. ξ. For k ∈ Λ± (η, δ) we consider the inhomogeneous Volterra Integral Equation [56] Z ∞ v± (ξ, k) = V± (ξ, k) − Kk (ξ, t)Fk (t)v± (t, k) dt (4.13) ξ

where Fk (t) = 12 Sφ(t, k) + µ is the function that expresses the difference between the Schr¨odinger equation (4.1) and the comparison equation (4.11) and K(ξ, t) is the Green’s function associated with equation (4.10): K(ξ, t) = W (V− , V+ )−1 {V+ (ξ)V− (t) − V+ (t)V− (ξ)}

(4.14)

(the parameter k being suppressed), with Wronskian W (V− , V+ ) := V− V+0 − V−0 V+ = 2ik. To give (4.13) meaning we need to check if the definition makes sense and a solution can be found. We explicitly compute Sφ and thus F using (4.12), obtaining
2 − 2k 4 cosh(2ξ) + Z sech(ξ) 12k 2 + 5Z sech(ξ) 10k 4 − Z+ + + Sφ (ξ) = 8 (Z+ + k 2 cosh(ξ))2 and thus, for real ξ and for every k ∈ Λ+ (η, δ), we have lim |F (ξ)| =

ξ→∞

1 8

+µ

CF := sup |F (ξ)| < ∞.

and

(4.15)

ξ∈(0,∞)

Of course CF depends on Z+ , µ and k, thus on η and δ. Moreover from (4.10) and (4.7), writing k ∈ Λ+ (η, δ) as k = kr + iki (kr , ki real), we get  φ(ξ,k)  √ −ξ ik
ξ k ≤ CV e− 2 exp − ki eξ (1 + o(1) , 2 |V± (ξ, k)| = 2e (1 + o(1)) e (4.16) 2

17



where CV (k) := sup

 eξ/2 |k/φ0 (ξ, k)| < ∞ by (4.8). Therefore for 0 < ξ ≤ t < ∞ we

ξ∈(0,∞)

have s 1   k2 |K(ξ, t)| = ei(φ(ξ,k)−φ(t,k)) − ei(φ(t,k)−φ(ξ,k)) 0 0 2ik φ (t, k)φ (ξ, k) q   2 Rt ξ+t C Z+ dτ , ≤ V e− 2 CK exp −ik ξ cosh(τ ) 1 + k2 cosh(τ ) 2 where

q  Rt CK (k) := sup 1 − exp 2ik ξ cosh(τ ) 1 + t,ξ∈R+

k2

Z+ cosh(τ )

(4.17)

 dτ ≤ 2.

It follows from (4.15), (4.16) and (4.17) that the Volterra Integral Equation (4.13) is well-defined as a mapping from the function space n o
C± (η, δ) := f ∈ C 2 (0, ∞)×Λ± (η, δ) ∀k ∈ Λ± (η, δ), kf kk := sup f (x, k) e∓iφ(x,k) < ∞ x∈(0,∞)

(4.18) to itself. In particular, being V± ∈ C± (η, δ) we can apply the Picard iteration procedure to find a solution of the equation and prove its existence. We claim that the solution must be unique. Suppose that there exists two solutions v+ , v˜+ ∈ C+ of (4.13), then Z ∞ ψ(ξ, k) := v+ (ξ, k) − v˜+ (ξ, k) = − K(ξ, t)F (t)ψ(t, k) dt. (4.19) ξ

At this stage, it is not obvious that the r.h.s. of (4.13) is a contraction, that would allow us to conclude the proof in a standard way. In the rest of the proof we show that for appropriate initial values this is indeed the case, therefore proving the unicity and the uniformity of the estimates. The previous estimates applied to (4.19) give Z ∞ Z ∞ |ψ(ξ, k)| = K(ξ, t)F (t)ψ(t, k) dt ≤ |K(ξ, t)F (t)ψ(t, k)| dt ξ ξ s Z s CK CF Cψ k iφ(ξ,k) ∞ k ≤ e dt φ0 (ξ, k) 2 φ0 (t, k) ξ Z CK CF Cψ CV − ξ iφ(ξ,k) ∞ √ − t e 2 e ≤ 2e 2 (1 + o(1)) dt 2 ξ Z CK CF Cψ CV CI − ξ iφ(ξ,k) ∞ − t −ξ iφ(ξ,k) 2 2 ≤ e e e dt = Cψ Ctot e e (4.20) 2 ξ q − 12 √  Z+ where Cψ (k) := kψkk , CI := supξ∈(0,∞) 2 (1 + e−2ξ ) 1 + k2 cosh(ξ) and Ctot := CK CF CV CI . Using equations (4.19) and (4.20) we can reiterate the procedure, in fact defining Z ∞ Z ∞ ψ1 (ξ, k) := K(ξ, t)F (t)ψ(t, k) dt and ψn (ξ, k) := K(ξ, t)F (t)ψn−1 (t, k) dt, ξ

ξ

18

one can prove by induction that n e−nξ C n e−nξ iφ(ξ,k) Ctot iφ(ξ,k) ≤ Cψ tot e e (2n − 1)(2n − 3) · · · 3 · 1 n! (4.21) uniformly in k ∈ Λ+ (η, δ) and for all n ∈ N. The convergence of |ψ(ξ, k)| = |ψn (ξ, k)| ≤

∞ X n=1

Cψ

  n Ctot −ξ e−nξ eiφ(ξ,k) = Cψ eiφ(ξ,k) eCtot e − 1 n!

(4.22)

implies that |ψ(ξ, k)| = 0, i.e. v˜+ = v+ . The same inequality implies that after some iterates the homogeneous integral equation is a contraction, and coupled with the bounds on V+ it implies that (4.13) has a unique fixed point. This proves the existence and uniqueness of the solution. In fact if we define Z ∞ K(ξ, t)F (t)vn−1,+ (t, k) dt, v0,+ (ξ, k) := V+ (ξ, k) , vn,+ (ξ, k) := − ξ

P∞

the Picard iteration converges to v+ = n=0 vn,+ , and the series converges absolutely −ξ uniformly in k ∈ Λ+ (η, δ) with |v+ (ξ, k)| ≤ |V+ (ξ, k)| eCe for some positive constant C. Therefore one has v+ (ξ, k) = V+ (ξ, k)(1 + o(1)) as ξ → ∞ and (4.9) holds. The fact that all the bounds are valid for k ∈ R completes the proof. Remark 4.7. It is possible to compute an explicit bound like (4.21) using the fact that C n e−nξ ξ |vn,+ (ξ)| ≤ CV e− 2 eiφ(ξ,k) totn . 2 n! In particular the dependence on µ, the parameter of the short-range potential in (4.3), appears in the constant Ctot . In view of the previous estimates it can be bounded by |µ|O(1). Therefore we can be more precise and estimate v± (ξ, k) =

√


ξ 2e− 2 e±iφ(ξ,k) 1 + M± (ξ, k, µ)

as

ξ → ∞,

−ξ

where for some constant C 6= 0 we have M± (ξ, k, µ) = eC|µ|e o(1).

(4.23) ♦

Remark 4.8. Let w be any other family of solutions of (4.1), analytic in k ∈ C \ {0} and satisfying for k ∈ Λ+ (η, δ) the estimate w(ξ, k) = o(1) as ξ → ∞. Then w(ξ, k) = γ(k)v+ (ξ, k), where γ(k) is a nowhere-vanishing analytic function of k ∈ Λ+ (η, δ).

19

♦

Remark 4.9. In case Z+ = 0, the solutions of (4.1) are given by linear combinations of the modified Mathieu functions (Mc and Sc) [18, §16.6]. In particular, if we look at their asymptotic behaviour, we find out that up to a constant factor   k2 k2 v+ (ξ, k) = Mc µ − , , ξ (4.24) 2 4 where Mc(a, q, x) is the modified Mathieu cosine, i.e. the solution of y 00 (x) − (a − 2q cosh(2x))y(x) = 0 √ that decays for q ∈ C+ . It is well-known [41, Chapter 2] that the function in the RHS of (4.24) admits an analytic continuation through the positive real axis on the negative complex plane for −π/2 ≤ arg(k) ≤ π/2 and that for x → ∞ and k ∈ C+ it has the following asymptotic behaviour [18, 41]  

x k2 k2 Mc µ − , , x = e− 2 exp i k2 ex (1 + o(1)) 1 + o(1) , 2 4 in line with the estimates (4.7) and (4.9), valid for all Z+ .

♦

For what follows we will need to work in a slightly different setting. If we perform the change of variable defined by ξ 7→ Log(x+1) (with the principal branch Log of the logarithm), for v˜(x, k) := v(Log(x + 1), k) Equation (4.1) takes the form
0 (x + 1)˜ v 0 (x, k) + h−2 q(x, k, Z+ , µ) v˜(x, k) = 0 with 2

Z+ µ k x + 1 + 2(x + 1)−1 + (x + 1)−3 + 1 + (x + 1)−2 − . q(x, k, Z+ , µ) := 4 2 x+1 (4.25) where x > 0, h > 0 and k ∈ C \ {0}. As before we assume h = 1 for the moment. Remark 4.10. In this case Theorem 4.5 and Remark 4.8 is still valid and in accord with the Liouville-Green Theorem we have two unique solutions that as x → ∞ are asymptotic to 1 e±iΨ(x,k) (1 + o(1)) x+1    1 =√ exp ±i k2 x + Z2k+ log(x + 1) + k2 x+1   2
 Z (1 + o(1)) · exp ±i 4k+3 (x + 1)−1 + O (x + 1)−2

v˜± (x, k) = √

(4.26)

where Ψ(x, k) = φ(Log(x+1), k). The asymptotic behaviour (4.26) holds uniformly for k in any sector Λ± (η, δ) = {k ∈ C | η ≤ arg(±k) ≤ π − η, |k| ≥ δ} with η ≥ 0 and δ > 0. The family of solutions defined by (4.26) is analytic in k ∈ C± \ {0} and extends continuously to k ∈ C± \ {0}. ♦ Remark 4.11. From now on we write with an abuse of notation φ(x, k) in place of Ψ(x, k). ♦

20

Before presenting Theorem 4.14, the main result of this section, we need the following lemma. Lemma 4.12. Let K be a compact set in C \ {0}. Then for any −π < θ < π, there is a constant Aθ such that any solution of Equation (4.25) verifies the estimate 1 |˜ v (x, k)| ≤ Aθ (|c| + |c0 |) √ e|=φ(x,k)| x

(4.27)

for x ∈ eiθ [0, ∞) and k ∈ K, where c = v˜(0, k), c0 = v˜0 (0, k) are the initial data at x = 0. Proof. We start proving (4.27) in the case η ≤ | arg k| ≤ π − η for any η ≥ 0 and θ = 0 (i.e. x ∈ (0, ∞)). All the constants that we are going to use without an explicit definition are defined as previously. Using the approximate solutions given by (4.10) defined by V± (x, k) := V± (log(x + 1), k), we determine a+ and a− from the initial data requiring c = a+ V+ (0, k) + a− V− (0, k),

0 0 , c0 = a+ V+ (0, k) + a− V− (0, k).

Then v˜(x, k) satisfies the Volterra Integral Equation Z x dt v˜(x, k) = a+ V+ (x, k) + a− V− (x, k) + K(x, t)F(t)˜ v (t, k) t+1 0

(4.28)

(4.29)

where K(x, t) := K(Log(x + 1), Log(t + 1)) and F(t) := F (Log(t + 1)) are defined from the respective function (4.14) and (4.13). Notice similarly as in the previous theorem that for 0 ≤ t ≤ x there exist constants C0 (η, δ) and CV such that we have
C0 (η, δ) 1 1 |K(x, t)| ≤ exp |=(φ(x, k) − φ(t, k))| 0 0 2 φ (x, k) φ (t, k) 2
C C0 (η, δ) 1 p ≤ V exp |=(φ(x, k) − φ(t, k))| . (4.30) 2 (x + 1)(t + 1) Define now V(x, k) =

√

2(|a+ | + |a− |) √


1 exp |=φ(x, k)| . x+1

(4.31)

The sequence Z v˜0 (x, k) := a+ V+ (x, k) + a− V− (x, k) ,

x

K(x, t)F(t)˜ vn−1 (t, k)

v˜n (x, k) := 0

dt , t+1

is uniformly convergent. In fact, suppressing the dependence of the constant on η and δ, we have |˜ v0 (x, k)| ≤ CV V(x, k) and, using the transformed version of (4.30), it follows by induction that 1 |vn (x, k)| ≤ V(x, k)Ln (x), (4.32) n! where Z x Z x 1 dt 1 dt 1 √ ≤ C0 CV CF √ L(x) := C0 |F(t)| φ0 (t, k) |F(t)| t + 1 = C0 CV t+1 t+1 x+1 0 0

21

P∞ is uniformly bounded for x ∈ (0, ∞). Therefore ˜n (x, k) converges uniformly and n=0 v absolutely and coincides with the given solution v˜(x, k) of (4.29) for η ≤ | arg k| ≤ π − η, η ≥ 0. In particular being a± bounded in terms of the initial data c and c0 , we obtain (4.27) for real values of x. At this point it is enough to notice that as soon as we do not cross the branch cut of the logarithm, all the inequalities and the equations written up to this point are valid, therefore the result holds replacing x with eiθ x for every −π < θ < π.

4.2. Analytic continuation We are ready to prove that the functions v± can be analytically extended in k up to the positive real axis. To this end we consider the transformed form v˜± . Remark 4.13. The potential q defined in (4.25) is analytic in C \ (−∞, −1]. Therefore its analyticity in the cone Σα,β := {z ∈ C \ {0} | −α < arg z < β} for all α, β ∈ [0, π) is clear.

(4.33) ♦

Theorem 4.14. Let v˜± (x, k) be defined as in Remark 4.10. Then v˜+ (x, k) admits an analytic continuation in k through the positive real k-axis into the region {k ∈ C \ {0} | −β < arg k < β} , v˜− (x, k) admits an analytic continuation into {k ∈ C \ {0} | −α < arg k < α} , for any α, β ∈ [0, π) and both verify the asymptotic relation (4.9) 1 v˜± (x, k) = √ e±iφ(x,k) (1 + o(1)) x

as x → ∞ in Σα,β ,

(4.34)

where (4.34) holds locally uniformly in k and uniformly in x. Furthermore an analytic continuation of v˜+ (x, k) and v˜− (x, k) through the negative real axis is defined via v˜+ (x, k) = v˜− (x, −k).

(4.35)

Remark 4.15. If α + β > π, the analytically continued function v˜± (x, k) may be doublevalued for k ∈ C∓ . By an abuse of notation we denote the corresponding, possibly not simply-connected, domain by D± (α, β) := {k ∈ C \ {0} | −β < arg(±k) < π + α} . See Figure 4.2.

22

(4.36)

0

-1

1

1

1

��(�)

D+ (α ,β )

0

0

D- (α ,β ) -1

-1 0

-1

1

��(�) Figure 4.2: Domains D± for α = 2π/3 and β = 2π/5.

Proof. It is well-known [12, Chapter 3.7] that, as solutions of the linear differential equation (4.25) with analytic coefficients, v˜± (x, k) admit an analytic continuation in x into the region Σα,β . The main point of this proof is to use this information to obtain the analyticity in k via dilation. More in details we will imitate the strategy of [3, Theorem 2.6], refining the crude bound of Theorem 4.12 by using the Phragmen-Lindel¨of principle. This allows us to identify the dilated solutions with a decaying solution of the dilated equation. In view of Lemma 4.12, (up to multiplication with a function only depending on k) this solution is uniquely defined by the asymptotic behaviour as x goes to infinity. Let us consider v˜+ (z, k) along a ray Γ := {z ∈ C \ {0} | arg z = γ} with 0 < γ < β. Then for x > 0 and k ∈ C+ \ {0}, the function ω(x, k, γ) := v˜+ (eiγ x, k)

(4.37)

satisfies the equation
0 e2iγ (eiγ x + 1)ω 0 (x, k) + 2 q(eiγ x, k, Z+ , µ) ω(x, k) = 0 h

(4.38)

with q from (4.25). Moreover the initial data 0 ω 0 (0, k, γ) = eiγ v˜+ (0, k),

ω(0, k, γ) = v˜+ (0, k),

(4.39)

are analytic in k ∈ C+ \ {0}. To obtain an analytic continuation of v˜+ (x, k) into the lower half-plane, first observe that by the Liouville-Green Theorem and Remark 4.10, Equation (4.38) has a unique solution ω+ (x, k, γ) in the cone −γ < arg k < π − γ characterized by the asymptotic relation ω+ (x, k, γ) = √

1 eiγ x

iγ x,k)

eiφ(e

23

(1 + o(1)) as x → ∞.

(4.40)

We claim that in fact for x ∈ (0, ∞),

ω+ (x, k, γ) = ω(x, k, γ)

0 < arg k < π − γ.

(4.41)

0 (0, k, γ) provide the analytic continuation of the initial data for Then ω+ (0, k, γ) and ω+ v˜+ (x, k) into the region −γ < arg k < 0, implying that v˜+ (x, k) can be continued analytically into the lower half-plane. To prove (4.41), we observe that x 7→ v˜+ (x, k) is of exponential type for x ∈ Σα,β and decays exponentially for =(k) > 0. Then it follows from the Phragmen-Lindel¨of principle [13, VI.4], applied to
√ g(x, k) := x exp − iφ(x, k) v˜+ (x, k) (4.42)

that for fixed =(k) > 0 the function v˜+ (x, k) decays exponentially as x → ∞ in a small cone containing (0, ∞). Therefore Remark 4.10 and Remark 4.8 applied to the dilated function ω+ (x, k, γ e) for some small γ e > 0 imply that ω+ (x, k, γ e) is a multiple of ω(x, k, γ e). This means moreover that it decays at a rate given by the expected function √

1 eieγ x


exp iφ(eieγ x, k) .

We can repeat this procedure a finite number of times and deduce that for fixed k the analytic function g(x, k) is uniformly bounded as x → ∞ within an angle − < arg x < γ + for some  > 0. Since by (4.26) lim g(x, k) = 1, x→∞

it follows from Montel’s theorem [13, VII.2] that this limit is assumed uniformly as x → ∞ in 0 ≤ arg x ≤ γ. This proves (4.41). Since γ ∈ (0, β) was arbitrary, we obtain an analytic continuation of v˜+ (x, k) to −β < arg k < π. It remains to prove (4.34). For −α < γ < β we can apply Lemma 4.12 to the dilated function ω(x, k, γ) to have g(x, k) = O(1) as

x → ∞ within Σα,β .

(4.43)

We already know from (4.41) that g(x, k) → 1 as x → ∞ along any ray such that 0 < η ≤ arg(kx) ≤ π − η for some η ≥ 0. Therefore we have that also locally uniformly in k ∈ C \ {0}, −β < arg k < π g(x, k) = O(1)

as x → ∞ within Σα,β

and g(x, k) is uniformly bounded along the boundary rays of Σα,β . That g(x, k) is uniformly bounded in x ∈ Σα,β is now a consequence of the Phragmen-Lindel¨of Principle. The fact that g(x, k) tends to 1 as x → ∞ since it does so along some ray contained in its interior, completes the proof of the theorem. Remark 4.16. The analytical extension of v˜(x, k) = v(Log(x + 1), k) gives in turn the extension of v(ξ, k). ♦

24

4.3. Generalised eigenfunctions, Green’s function and the scattering matrix We are now ready to construct the main elements for the partial wave expansion required to give a definition of the resonances of our operator. We considered in the previous section the outgoing respectively incoming solutions as the solutions meeting a “regular” boundary condition at infinity. Because of the fact that the boundary conditions are at infinity it requires some work to prove that they can be analytically extended to the second Riemann sheet across the positive real axis. This is much simpler for the solution v˜0 (x, k) of (4.25) (or the corresponding v0 (ξ, k) of (4.1)) that is regular in 0 in the sense of the boundary conditions derived from (2.16), i.e. v˜0 (0, k) = 1 ,

v˜00 (0, k) = 0.

(4.44)

Being the solution of a boundary problem with analytic coefficients and analytic initial conditions, the following theorem follows as a corollary of the standard theory of complex ordinary differential equations (see [12, Chapter 1.8]). Theorem 4.17 (The regular solution). The unique solution v˜0 (x, k) of (4.25) defined by the condition (4.44) is analytic in the cone x ∈ Σα,β , k ∈ C \ {0} defined in (4.33) and satisfies v˜0 (x, k) = v˜0 (x, −k). (4.45) Remark 4.18. Working with (4.25) or (4.1) is equivalent. We will use each time the representation that makes the proofs and the computations easier. Therefore in what follows we do not continue to remark that the properties are equivalent. It is always possible to understand in which setting we are working, looking at the name of the functions and the variables. From now on, we will always assume that the Wronskian is defined in its generalised form given by
Wx (f, g) := p(x) f (x)g 0 (x) − f 0 (x)g(x) , where the notation comes from (A.6). We are finally ready to introduce the basic elements for scattering theory on the half-line. We call Jost functions associated to the radial equation (4.25) and our choice of phase function φ(x, k) the Wronskians
f± (k) := W v˜± (•, k), v˜0 (•, k) . (4.46) They connect the regular solution to the incoming and outgoing ones via the identity W (˜ v− , v˜+ )˜ v0 = f+ v˜− − f− v˜+ ,

with W (˜ v+ , v˜− ) = 2ik,

(4.47)

that follows expanding explicitly the Wronskian and using the asymptotic behaviour of the solutions in their domain of analyticity. In particular this implies the following corollary of Theorem 4.17 and Theorem 4.14.

25

Corollary 4.19. The Jost functions f± (k) are analytic in k ∈ D± (α, β) defined in (4.36) and verify
√ f± (k) = ±(2ik) lim eiγ/2 x exp ±iφ(eiγ x, k) v˜0 (eiγ x, k), (4.48) x→∞

where γ ∈ (−α, β) satisfies γ ≷ − arg(k) according to the choice of sign of (4.48). It will be convenient for what follows to change the normalisation v˜0 (0, k) = 1 to one at “infinity” in the sense of Corollary 4.19. Namely if f+ (k) 6= 0, we define the generalised eigenfunction of the radial equation (4.25) and our choice of phase function φ(x, k) the function e(x, k) := f+ (k)−1 v˜0 (x, k). (4.49) With this notation we introduce for k ∈ Σα,β with f+ (k) 6= 0 the radial Green’s function G(x, x0 ; k) := e(x< , k)˜ v+ (x> , k),

(4.50)

where for x, x0 > 0, x< := min{x, x0 } and x> := max{x, x0 }. G(x, x0 ; k) is a fundamental solution of the radial Schr¨ odinger equation (4.25). Remark 4.20. We now consider the spectral parameter µ appearing in Equation (4.1) as a perturbation of the operator Kξ defined in (2.14). Consequently we will write Kξ (Z+ , µ) := Kξ + µ for the perturbed operator.

♦

Remark 4.21. Notice that eventual zeros of f+ (k) for k ∈ C+ \{0} correspond to eigenvalues of the operator. ♦ In view of Theorem 4.14 and 4.17, G(x, x0 ; k) possesses a meromorphic continuation in k into the possibly two-sheeted domain, projecting to D+ (α, β) defined by (4.36). Finally we introduce the so-called scattering matrix element s(k) =

f− (k) f+ (k)

(4.51)

which in view of Corollary 4.19 is a meromorphic function of k over D+ (α, β) ∩ D− (α, β). Lemma 4.22. Let x, x0 > 0 and −β < arg(k) < α. 1. The radial Green’s function and the radial generalised eigenfunctions satisfy the functional relation G(x, x0 ; k) − G(x, x0 ; −k) = −2ik e(x< , k)e(x> , −k).

(4.52)

2. The scattering matrix element satisfies the following relation s(−k) = s(k)−1 .

(4.53)

3. The scattering matrix elements and the radial generalised eigenfunctions satisfy the functional relation s(k)e(x, −k) = e(x, k). (4.54)

26

Proof. From (4.35) and (4.45) we have that

f+ (−k) = W v˜+ (•, −k), v˜0 (•, −k) = W v˜− (•, k), v˜0 (•, k) = f− (k)

(4.55)

for k ∈ D+ (α, β) ∩ D− (α, β). Therefore, using (4.47) and the definitions of the radial Green’s function and the radial generalised eigenfunctions, we get G(x, x0 ; k) − G(x, x0 ; −k) = e(x< , k)˜ v+ (x> , k) − e(x< , −k)˜ v+ (x> , −k)
−1 −1 = v˜0 (x< , k) f− (−k) v˜+ (x> , k) − f+ (−k) v˜+ (x> , −k) = v˜0 (x< , k)f− (−k)−1 f+ (−k)−1 f+ (−k)˜ v+ (x> , k) − f− (−k)˜ v+ (x> , −k)




= v˜0 (x< , k)f+ (k)−1 f+ (−k)−1 f+ (−k)˜ v− (x> , −k) − f− (−k)˜ v+ (x> , −k)




= −2ik e(x< , k)f+ (−k)−1 v˜0 (x> , −k) = −2ik e(x< , k)e(x> , −k). The second part and the third part follows as a direct application of (4.55) to the definition of the scattering matrix elements. A first consequence of Lemma 4.22 is that it is enough to discuss the scattering matrix elements in the angle −β < arg(k) < α. With the above definitions we can discuss the notion of eigenvalues for the radial nonselfadjoint Schr¨ odinger operator Kξ (Z+ , µ) in L2 ((0, ∞), cosh2 (ξ)dξ). We define n o EZ+ ,µ := k ∈ C+ \ {0} | f+ (k) = 0, e−ξ/2 eiφ(ξ,k) ∈ L2 ((0, ∞), cosh2 (ξ)dξ) . (4.56) If k ∈ EZ+ ,µ , we call k an eigenvalue of this quadratic eigenvalue problem. All other zeros of the Jost function f+ (k) are called resonances of Kξ (Z+ , µ) and we denote them by  RZ+ ,µ := k ∈ D+ (α, β) \ EZ+ ,µ | f+ (k) = 0 . (4.57) ξ

Remarks 4.23. 1. The condition ξ 7→ e− 2 eiφ(ξ,k) ∈ L2 ((0, ∞), cosh2 (ξ)dξ) is automatically fulfilled when k ∈ C+ \ {0}, independently of µ. 2. There cannot be real positive k ∈ EZ+ ,µ . In fact, if there would exist k ∈ (0, ∞) in EZ+ ,µ , then by Theorem 4.14 we would have v+ (ξ, k) ∈ L2 ((0, ∞), cosh2 (ξ)dξ), but it is evident from the asymptotic behaviour of v+ that this is impossible. On the other hand, we cannot exclude a priori the presence of real k in RZ+ ,µ . 3. Two Jost functions cannot vanish simultaneously in −β < arg(k) < α, otherwise v˜+ and v˜− (or v+ and v− ) would be linearly dependent in contradiction with their asymptotic behaviour. Therefore the points of EZ+ ,µ ∪ RZ+ ,µ contained in −β < arg(k) < α are in one to one correspondence with all the poles of the scattering matrix elements s(k). In view of the definitions (4.49) and (4.50), the set EZ+ ,µ ∪ RZ+ ,µ can be identified with the set of poles of the radial Green’s function G(ξ, ξ 0 ; k) or with the set of poles of the generalised radial eigenfunctions e(x, k). 4. The set RZ+ ,µ of resonances does not depend on the choice of the phase function which determines the Jost functions f± (k), the generalised radial eigenfunctions and the scattering matrix elements. ♦

27

5. Formal partial wave expansion of the Green’s function For real E we know from Remark 3.1 that the spectrum of Kη = Kη (E, Z− , h) consists of an infinite number of simple eigenvalues µ0 (E) < µ1 (E) < µ2 (E) < µ3 (E) < . . . tending to infinity, where in the notation of Remark 3.1 we have µn := λn +γ2 . These extend to analytic functions of E in some neighborhood of the real line. We shall denote by ϕn,E the eigenfunctions Kη (E)ϕn,E (η) = µn (E)ϕn,E (η), n ∈ N0 , normalised by 2

Z

+π

|ϕn,E (η)|2 dη = 1

kϕn,E k = −π

for E ∈ (0, ∞) and then extended analytically. We choose ϕn,E real for E real. Define K := F HG

(5.1)

with HG from Proposition 2.11 and F from (2.4). Instead of solving (HG − E)u = f in L2 (M, F (ξ, η)dξdη) for E ∈ C \ σ(HG ), we look at the solutions of
K − F (ξ, η)E u(ξ, η) = F (ξ, η)f (ξ, η). (5.2) We already know (see (2.14)) that
K − F (ξ, η)E u(ξ, η) = KE u(ξ, η) = (Kξ + Kη )u(ξ, η). Now, using the completeness of the orthonormal base {ϕn,E }n∈N for E ∈ R, u possesses the expansion X u(ξ, η) = un (ξ, η) with un (ξ, η) := ϕn,E (η)ψn,E (ξ), (5.3) n∈N0

where Z

+π

ψn,E (ξ) =

ϕn,E (η)u(ξ, η) dη. −π

This expansion extends to complex values of E by analyticity (note that no complex conjugate is involved, since ϕn,E is chosen real for E ∈ R). Analogously we get Z +π X F (ξ, η)f (ξ, η) = ϕn,E (η)gn,E (ξ) with gn,E (ξ) := ϕn,E (η)(F f )(ξ, η) dη. −π

n∈N0

(5.4) Substituting (5.3) and (5.4) into (5.2) one gets X X (Kξ + Kη ) un (ξ, η) = ϕn,E (η)gn,E (ξ) n∈N0

n∈N0

28

or equivalently X


ϕn,E (η) (Kξ (E) + µn (E)) ψn,E (ξ) − gn,E (ξ) = 0.

(5.5)

n∈N0

Remark 5.1. (5.5) extends to complex points E 6∈ σ(H), where Kξ (E) + µn (E) possesses an inverse Rn (E) by means of the Green’s function defined in (4.50). ♦ Z ψn,E (ξ) = Rn (E)gn,E (ξ) =

˜ E) Gn (ξ, ξ;

Z

+π

−π

(0,∞)

˜ ηe) de ˜ ϕn,E (e η )(F f )(ξ, η dξ,

(5.6)

using (5.5). Combining (5.6) and (5.3) we obtain ZZ X ˜ E)ϕn,E (e ˜ ηe) dξ˜ de Gn (ξ, ξ; η )(F f )(ξ, η ϕn,E (η) u(ξ, η) = M0

n∈N0

and we read off the partial wave expansion for the Green’s function X ˜ ηe; E) = ˜ E)(cosh2 ξ˜ − cos2 ηe). G(ξ, η; ξ, ϕn,E (η)ϕn,E (e η )Gn (ξ, ξ;

(5.7)

n∈N0

It would be of great interest to be able to prove that the sum converges in the sense of distributions in the product space D0 (M ) ⊗ D0 (M ). Then we could use our results on the analytic continuation of the Gn and of the angular eigenfunctions to give a meromorphic ˜ η˜; E) in E to the second Riemann sheet (or k ∈ C− ). continuation of the G(ξ, η; ξ, Anyhow, for each fixed N ∈ N, we can consider the restriction KN of the operator K to the subspace ΥN (E) :=

N M

Φn (E) ⊗ L2 ((0, ∞), cosh2 (ξ)dξ) ⊂ L2 ([−π, π], dη) ⊗ L2 ((0, ∞), cosh2 (ξ)dξ)

n=0

(5.8) where Φn (E) is the subspace spanned by ϕn,E . The relative Green’s function ˜ ηe; E) = GN (ξ, η; ξ,

N X

˜ E)(cosh2 ξ˜ − cos2 ηe) ϕn,E (η)ϕn,E (e η )Gn (ξ, ξ;

n=0

is the truncated sum obtained from (5.7). Being a finite sum of well-defined terms, it is convergent. Moreover it follows from the results of the previous sections that it possesses a meromorphic continuation in E to the second Riemann sheet.

6. Resonances for the two-centers problem With the expansion of Section 5 and the theory developed in the previous sections, we are finally ready to define the resonances for the two-centers problem and analyse some of their properties. This is done in Section 6.1.

29

The rest of the section is then devoted to asymptotically locate these resonances. In particular in Section 6.2 we show that the resonances can be computed as roots of some explicit asymptotic equation, and in the subsequent sections we explicitly solve this equation in different semiclassical energy regimes.

6.1. Definition of the resonances The operator Kη defined by (3.5) has discrete spectrum µn (k 2 ) admitting an analytic continuation in k 2 := E in some neighborhood of the real axis. At the same time for each µ, the resolvent of the operator Kξ (µ, Z+ ) (see Remark 5.1) can be extended in terms of k to the negative complex plane, having there a discrete set of poles km (µ). With the definitions given in Section 4.3 we set  2 En := k ∈ C+ \ {0} | f+ (k, µn (k 2 )) = 0, e−ξ/2 eiφ(ξ,k,µn (k )) ∈ L2 ((0, ∞), cosh2 (ξ)dξ) . (6.1) If k ∈ En (for some n ∈ N0 ), we call k an eigenvalue of the quadratic eigenvalue problem for K = K(Z− , Z+ ) defined in (5.1). All other zeros of the Jost function f+ (k, µn (k)) are called resonances of K(Z− , Z+ ) and we denote them by  Rn := k ∈ D+ (α, β) \ En | f+ (k, µn (k 2 )) = 0 . (6.2) Proposition 6.1. The sets En and Rn are made by an at most countable number of elements 2 )) = 0. km ∈ D+ (α, β) (m ∈ I ⊆ N) of finite multiplicity such that f+ (km , µn (km Proof. f+ (k) and µn (k 2 ) being non-constant analytic functions of k, the statement is clear. Remark 6.2. Notice that if k 2 is an eigenvalue of the full operator K (or its restriction KN ), then it must be an eigenvalue of Kξ (Z+ , µn ) for some µn (k 2 ) (i.e. an element of En ). ♦ Remark 6.3. By definition En ∩ Rn = ∅. Furthermore, it is clear looking at the asymptotic behaviour (4.7) of the phase function that it is impossible that k ∈ En and k ∈ Rn0 for n 6= n0 . ♦ Relying on the previous discussion and on Remark 4.23.2 we can switch from the k 2 plane to the k plane and refer to E N :=

N [

RN :=

En ,

n=0

N [

Rn

(6.3)

n=0

as the sets of eigenvalues and resonances of KN . Moreover, in view of Remark 4.23.2, the points of E N ∪ RN contained in D+ (α, β) ∩ D− (α, β) are in one-to-one correspondence with the poles of the scattering matrix elements sn (k) := s(k, µn ) and with the poles of the e k) := G(ξ, ξ; e k, µn (k)) for n ∈ {0, . . . , N }. Green’s functions Gn (ξ, ξ;

30

Remark 6.4. If we suppose that (5.7) is convergent, we can refer to E :=

∞ [

En ,

R :=

n=0

∞ [

Rn

(6.4)

n=0

as the sets of eigenvalues and resonances of K. As for the restricted operator, in view of Remark 4.23.2, the points of E ∪ R contained in D+ (α, β) ∩ D− (α, β) are in one-to-one correspondence with the poles of the scattering matrix elements sn (k) and with the poles of e k). the Green’s functions Gn (ξ, ξ; ♦

6.2. Computation of the resonances of Kξ Consider the equation 0 = Kξ (h)ψ(ξ) = −h2 ∂ξ2 ψ(ξ) − Z+ cosh(ξ)ψ(ξ) − E cosh2 (ξ)ψ(ξ)

(6.5)

with the condition ψ 0 (0) = 0. The potential V (ξ; Z+ , E) := −Z+ cosh(ξ) − E cosh2 (ξ) has a Taylor expansion around ξ = 0 given by  E 2 Z+  ξ e + e−ξ − eξ + e−ξ 2 4  Z+ ξ 2 + O(ξ 4 ) = A − ω 2 ξ 2 + O(ξ 4 ), = −Z+ − E − E + 2 q where A := −Z+ − E and ω = E + Z2+ . V (ξ; Z+ , E) = −

Let now E + Z2+ > 0. We would like to apply the theory developed in [6, 7, 8] and [50] to get the resonances from the eigenvalues en (h) = h(2n + 1)ω

(n ∈ N0 )

of the harmonic oscillator Hosc = −h2 ∂ξ2 + ω 2 ξ 2 , according to An (h, E, Z+ ) = −Z+ − E − ih(2n + 1)ω + O(h3/2 ). Remark 6.5. [6, 7, 8] and [50] are not directly applicable, as there it is essential to assume that the potential is bounded, and this is clearly false in (6.5). ♦ The problem stressed by the previous remark can be solved. With the change of variable given by y := sinh(ξ) : (0, ∞) → (0, ∞) we change the measure from cosh2 (ξ) dξ to p y 2 + 1 dy. At the same time the differential equation of Kξ (Z+ , µ) takes the form   p −h2 (y 2 + 1)∂y2 u(y) − h2 y∂y u(y) + µ − k 2 (y 2 + 1) − Z+ y 2 + 1 u(y) = 0.

31

Note that µ will correspond to an eigenvalue of the angular equation Kη , and as such it will be an analytic function of E. Moreover it will be real for real values of E (see Section 3). With the ansatz 1 u(y) := p v(y) 4 2 y +1 we can rewrite the differential equation in Liouville normal form as
y2 + 1 2 2 p −h ∂ v(y) + V (k, Z , µ, h; y)v(y) =0 + y 4 y2 + 1 where

(6.6)

Z+ µ y2 − 2 V (k, Z+ , µ, h; y) := −k 2 − p + − h2 . y 2 + 1 1 + y 2 4(y 2 + 1)2

This potential V has the following properties: • it is smooth in (0, ∞); • it is bounded; • it is analytic in a cone centered at the positive real axis; • it has a non-degenerate global maximum at y = 0; • around the maximum V can be expanded in Taylor series as V (k, Z+ , µ, h; y) = A − ω 2 y 2 + O(y 4 ), q 2 where A := −Z+ − k 2 + µ − h2 and ω = µ + 45 h2 − Z2+ . Therefore it satisfies the assumptions of [6, 7, 8] and [50], there a resonance is an exact zero of some symbol in the semi-classical parameter, and we are left to compute the leading terms of this symbol. This allows us to approximate the resonances with the eigenvalues of the harmonic oscillator according to An (h, E, Z+ , µ) = −Z+ − k 2 + µ − ih(2n + 1)ω + O(h3/2 ).

(6.7)

This given, we have a solution of (6.6) if v is identically 0 or if An = 0. In summary, Proposition 6.6. For any given Z+ and µ, the resonances of Kξ (Z+ , µ) are asymptotically given by the zeroes of a symbol An (h, E, Z+ , µ) whose expansion as h → 0 is provided by (6.7). From this formula one can have a first very rough approximation of the resonances En = kn2 in orders of

Z− 2

µ ehn

> 0. Define

Z2 := − − + 4E

s  2  Z− E 1− (2n + 1) h. 4E 2

(6.17)

h en − µhn = O(h3/2 ). There exists an eigenvalue µhn of Kη and a constant c such that µ  h  Moreover, the interval µ en − 2ch3/2 , µ ehn + 2ch3/2 contains at least two eigenvalues of Kη . Remark 6.16. It can be proved by standard methods involving the IMS formula [14, Chapter 3.1] and Agmon estimates [2] that the distance between the eigenvalues in each pair is of the order exp(−C/h) with C ∈ (0, ∞). ♦ We can use this result in combination with (6.7). Proposition 6.17. The resonances in the set Rn ∩ { Z2− > 0} (see (6.2)) are given asymptotically as h → 0 by the solutions of the following equation An (h, E, Z+ , µ+ m (h, E)) = 0. Neglecting the error terms, the resonances for Z2 −E−Z+ − 4E−

q + E−

2 Z− 4E

rq (2m + 1)h+ih(2n+1)

Z− 2

E−

(6.18)

> 0 are given by the solutions of 2 Z− 4E

(2m + 1) h −

2 Z− 4E

−

Z+ 2

= 0.

Remark 6.18. For Z− = 0 we recover (6.9) of the previous section. On the other hand, in Section 6.3 the approximation error is of order O(h2 ) instead of O(h3/2 ). ♦ For 0 < E < Z2− the bottom of the potential is reached at π and thus we have to expand the potential around this other point. It turns out that in this case the eigenvalues are approximated by q µ bhn := E − Z− +

Z− 2

37

− E (2n + 1) h.

(6.19)

Proposition 6.19. The resonances in the set Rn ∩ {0 <